Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra

TL;DR

This paper reveals that 4x4 nilpotent matrices encoding the Schrödinger equation naturally organize into structures resembling grand unified theories, providing insights into fermion families and gauge symmetries.

Contribution

It rigorously identifies and classifies nilpotent matrices that encode quantum dynamics, uncovering their group-theoretic structures related to grand unified theories.

Findings

Matrices form three groups of sixteen with distinct scattering properties

Symmetric sector mirrors quark-lepton decomposition of Pati-Salam model

Results suggest 4x4 algebra as a shadow of SO(10) GUT embedding

Abstract

We show that nilpotent matrices that yield the Schrodinger equation from its first order form encode the fingerprints of grand unified theories. We perform a rigorous search for all such nilpotent matrices and find that the resulting matrices naturally organize into suggestive group theoretic structures without any other a priori assumptions. The antisymmetric sector consists of three groups of sixteen matrices, each of which further splits as 16 = 12 + 4 and exhibits unique characteristics in the step potential scattering problem. The symmetric zero-diagonal sector also forms three families, mirroring the quark-lepton decomposition of the Pati-Salam model. These results may help answer why there are three families of fermions and also demonstrate that the 4 x 4 matrix algebra is a compact, nontrivial shadow of the SO(10) embedding, with fermion-like and gauge-like subspaces.